MODELING IMPERFECT KNOWLEDGE IN MANAGEMENT AND INSURANCE

RISK

by Mark Jablonowski

Abstract: The author examines the problems for risk managers associated with knowledge imperfections. He describes how fuzzy logic can be used to deal with situations when there is a lack of knowledge

INTRODUCTION

Key to analysis and decision mak- ing in insurance, risk management and allied fields is an understanding of the form of uncertainty attributed to ran- domness. This type of uncertainty is specified using probabilities. Less rec- ognized, but equally important, is the uncertainty that results from knowl- edge impegection. Knowledge imper- fection is a natural concomitant of complex and dynamic environments. It is reflected in imprecision, vague- ness and approximation.

Knowledge imperfection manifests itself in our inability to distinguish among alternatives. Under knowledge imperfection, we can only specify model parameters and measurements as a range, or interval, of possibilities. Usually we can associate various de- grees of confidence with the bounds of these intervals. This “graded nonspecificity” is known as fuzziness.

Through the formal study of fuzzi- ness we are better able to understand the processes that underlie decisions in risk management and insurance. This understanding, in turn, leads to better decisions. A rigorous framework for the analysis of fuzziness also al- lows us to automate the reasoning pro- cess, further improving our ability to make sound decisions in a “real world’ setting .

FORMALIZING KNOWLEDGE IMPERFECTION

The type of uncertainty attributable to fuzziness can be formally repre- sented using membership functions. These show the range of plausible al- ternatives, graded by confidence. Membership is usually measured on a continuous scale that goes from 0 to 1.

A membership of 1 indicates com- plete confidence in an alternative’s being a member of the set of plausible alternatives, while a membership of 0 means no confidence. Perfect knowl- edge is represented by a single alter- native having a membership of I, and all others 0. Complete ignorance is defined by our inability to distinguish

Mark Jablonowski. ARM, CPCU. is risk manager for the Hamilton Standard division of United Technolo- gies Corporation in Windsor Locks, Connecticut.

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1.0 -

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0

among any alternatives to any degree. In this case, all alternatives show a membership of 1.

In real life, our knowledge usually falls somewhere between perfect knowledge and complete ignorance. Say that I have just come in from out- doors, and a friend asks me about the temperature. Not having access to a thermometer, my knowledge is lim- ited. Based on my experience, I remark that it is “cool”.

Figure 1 shows plausible precise and fuzzy membership functions that

Fuzzy

I I I 1 I I I I 1

define the word cool, as i t pertains to springtime air temperatures in New England. The precise definition im- plies very exact information, such as might be obtained from a thermom- eter reading or recent weather report. The fuzzy definition, on the other hand, suggests a range of possibilities. The most representative are those tempera- tures between 45 and 55 degrees. Tem- peratures below 35 degrees, and above 65 are clearly not what we would call cool. Borderline cases (e.g., 40 de- grees) are assigned partial member-

1 .o

a .- c 3 .5

2

0

Figure 1 Defining Cool

30 35 40 45 50 55 60 65 70 Temperature (“F)

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Modeling Imperfect Knowledge in Risk Management and Insurance

ships, suggesting that they simply can not be unambiguously classified.

Though imperfect, fuzzy statements can convey useful information about the properties of the world. Knowing only that it is cool outside, you would probably not wear shorts that day. The amount of uncertainty communicated can also influence decision making.

Perhaps, unbeknownst to me, my friend was going to perform a scien- tific experiment that required an air temperature of exactly 50 degrees. Had I communicated the temperature as 50 degrees, when 45 degrees was at least a possibility, my friend would have been misinformed (with significant consequences). Receiving only the in- formation that it is cool, she would have perhaps sought out a thermom- eter.

Fuzzy membership functions are usually obtained by direct questioning of human subjects. For example, we could ask an insurance underwriter to specify the degree to which various loss ratios fit the concept low loss ru- tio for a particular line of business.

Though obtained from the experi- ence of the individual, memberships are not “subjective” in the sense of being peculiar to that individual. They do have a link to objective reality. This link is instrumental, in the sense we judge membership functions by how well they let us reason about a com- plex and uncertain world. In the de- velopment of membership functions, there is always a trade-off between specificity (and hence informativeness) and truth. By making specific state- ments under conditions of uncertainty, we risk that they may be false. On the other hand, nonspecific statements

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may become vacuous (e.g., “the tem- perature tomorrow will be between - 50 and 150 degrees”). Fuzzy sets are constructed so as to strike a balance between specificity and truth, based on available information.

Methods also exist for the auto- matic generation of fuzzy membership functions. For example, fuzzy classi- fication systems have been constructed using optimization techniques. Based on limited information and an appro- priate goal structure (specificity vis a vis truth), they are able to produce fuzzy representations that are consis- tent with human intuition. Represen- tations of imperfect knowledge can therefore be constructed using compu- tational, as well as biological, intelli- gence.

FUZZY REASONING

In insurance and risk management, we often reason using fuzzy concepts. For example, expert risk managers tell us that good deductibles should have a low probability of being exceeded. By insuring losses that occur too fre- quently, we end up with the uneco- nomical “trading” of losses (claims) for losses plus insurer expenses (pre- miums). Insurance is best used for in- frequent losses that can have a signifi- cant financial impact. However, the cutoff between frequent and infrequent losses (as reflected in their probability of exceedance) can not be specified exactly. The process is complicated by too many variables to permit a precise specification. For example, in addition to the financial effects of the deduct- ible on the organization, the risk man- agers must anticipate the reaction of

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the insurer. This all means that con- siderable fuzziness surrounds the choice of deductibles.

We can formalize this fuzziness us- ing membership functions. Figure 2 shows a membership for low probabil- iry ofpenetration, as it applies to de- ductible selection. It was obtained through expert interviews, reviews of the literature and a review of expert decisions in the field. When applied to the frequency of losses expected by an individual or organization decision, the function indicates a fuzzy range of deductible alternatives of varying “goodness.” Note that the fuzzy model implies that we cannot identify ideal deductibles: Only “good” and “less good” ones.

This application shows that, in prac- tice, the two types of uncertainty (ran- domness and fuzziness) can co-exist. We use fuzzy probabiliries when we cannot measure or model probabilities exactly. Usually, these fuzzy probabili- ties are expressed using words, such as “low probability”, “unlikely” or “slight chance”. The use of linguistic probabilities is, therefore, a quite natu-

ral response to knowledge imperfec- tion.

It should be obvious from the pre- ceding discussion that a recognition of the fuzzy nature of the world is nor an invitation to sloppy thinking. If good information is available, it behooves us to use it. However, intellectual hon- esty demands that we not construct models, or proffer measurements, to a level of precision that simply can not be justified. A reasonable creed for both the researcher and practitioner is this: Work to reduce knowledge im- perfection wherever possible, but re- alize that this reduction can have very real bounds. These bounds are deter- mined by the complexity and dynam- ics of the domain. Real wisdom is knowing when these bounds have been reached.

AUTOMATING THE REASONING PROCESS

The fuzzy representation of uncer- tainty is easily incorporated into com- puter systems that emulate the thought processes of human experts. These so-

Figure 2 Low Probability

0 .O1 .02 .03 .04 .05 .06 .07 .08 .09 . 10 Probability

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Modelinx Impeiject Knodrclge i n Risk Mnncrgemenr crnd Insitrance

called expert svstems provide a host of benefits, including the improvement of our understanding of expert reason- ing. They also permit us to share ex- pertise with a wide audience, includ- ing those who may have no “human” source of expertise to rely on when making critical decisions. There exist a variety of expert systems applications in science, business and industry. These range from the evaluation of credit applications to the control of complex industrial processes.

A simple expert system that formal- izes the intuitive process of deductible choice in organizational risk manage- ment is the Retention Evaluation Pro- gram (REP). The REP is based on the fuzzy model of retention selection de- scribed above. The membership func- tion for “low probability of penetra- tion” (Figure 2) is encoded into the program. The user inputs for the ex- posure under study the largest loss (for “per occurrence” deductibles) or the total loss (for “aggregates”) for each of the last ten years. The program then fits a probability of loss distribution to this data. This probability distribu- tion is matched to the penetration prob- ability membership function to deter- mine the fuzzy dividing line between frequent and infrequent losses. The result is a membership function that defines the fuzzy set of “good reten- tion points.” Evaluations of current or potential deductible selections are made accordingly.

The Retention Evaluation Program has been employed in a variety of en- vironments, including practical appli- cations in day-today risk management decision making, and as an educational tool. While its ability to make deci-

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sions “as good” as its human counter- parts may be arguable, the program offers insights into expert decision making process which are not possible using either unduly precise mathemati- cal formalizations, or purely anecdotal descriptions. The reasoning process embodied in the system is transparent, being fully contained in the program’s coding (currently in BASIC). The code is easily accessed and modified, facili- tating continuing research and devel- opment.

Despite their obvious usefulness in formalizing expert knowledge, impedi- ments to expert system construction exist. Among them, the difficulties that arise from attempting to extract an ac- curate description of problem solving methods from the human experts them- selves. Often this knowledge is deep seated, or “intuitive.” Getting experts to “think out loud” about the way they make decisions can be painstaking and time consuming. Even then, there is no guarantee that these introspective verbalizations reflect the underlying thought process. The use of traditional methods of programming can also make expert system production diffi- cult. Attempting to encode expert thought “rule-by-rule’’ using proce- dural programming languages is itself a grueling task. Subsequent implemen- tation is complicated by slow process- ing speeds.

Recent developments in computer and cognitive sciences alleviate some of the traditional bottlenecks of expert systems design and deployment. The most significant of these is the artifi- cial neural network, or ANN. ANNs are computer algorithms designed to mimic the functioning of the human

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brain. These networks have the capa- bility of learning complex input-out- put relationships from data. They model these relationships through nu- merically weighted connections among “nodes,” whose operation is similar to the neurons in the human brain. Differential activation of these nodes results in the expression of vari- ous patterns in the data. Connection weights are themselves “learned” through iterative training on input-out- put exemplars. Their highly parallel structure permits all-at-once process- ing of information, rather than the one- at-a-time (serial) processing of proce- dural programming languages. This greatly increases computational speed.

Their ability to express complex in- put -output re 1 at i o n s h i p s ( “ r u 1 e s” ) makes ANNs an ideal processing mechanism for expert systems. Net- works can be trained using sample in- put data along with expert responses (“outputs”). This at once speeds the knowledge acquisition process and helps us gain a more accurate picture of expert reasoning, unclouded by ex- pert “interpretation.” In effect, the expert’s actions speak for themselves.

The speed bottleneck is also allevi- ated by the ANN’S parallel architec- ture. This permits rapid development and deployment of relatively complex systems.

ANNs provide an environment for the v e r y natural integration of fuzziness and expert systems. Fuzzy membership functions can be used at both the input and output stages of the process. ANNs that have a neuronal structure that performs fuzzy reason-

ing directly have also been developed. While the introduction of fuzziness requires greater computational effort, this increased effort is counterbalanced by the increase in available process- ing speed made possible through the use of ANNs.

As a simple example of the appli- cation of an ANN-based fuzzy expert system (FES) we consider the expert analysis of failure probability of struc- tures. The failure of safety-critical structures like dams, bridges and build- ings is obviously of crucial concern to risk managers, insurers and society at large. A FES can be used to relate vari- ous factors, such as age of the struc- ture and its physical condition, to fuzzy failure probabilities. These probabili- ties will be fuzzy since statistical data on catastrophic failures is usually scarce.

“Expert opinion” is often used to assess these probabilities. While use- ful, these estimates are necessarily im- perfect. We can capture this knowledge imperfection using fuzzy membership functions.

We can train an ANN-based FES using input and output pairs consist- ing of, respectively, data on influen- tial factors (age, condition and so on) and the resulting fuzzy membership for failure probability. For ease of process- ing, the universe of relevant probabilities can be dimtized into informative land- mark values (e.g., lo4, lo-’). Upon training, the FES is able to re- late input data to fuzzy probabilities. The program is ready for use as a de- cision aid. Validation and testing pro- ceeds as with traditional expert sys- tems.

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Figure 3 shows the output screen of a simple prototype ANN-based FES, designed to assess the fuzzy probabil- ity of dam failure. Factors used for this prototype were age, condition and con- struction. Age is given in years since “first fill”, condition is rated subjec- tively on a scale of poor (1) to excel- lent ( 5 ) and construction is noted as either concrete (1 ) or earthen (2). An ANN was trained using hypothetical expert-generated data. Examples pre- sented to the network consisted of combinations of age, condition and construction information and the expert’s fuzzy probability responses based on this information. ANNs show a remarkable ability to generalize com- plex nonlinear relationships in data. This permits the FES to respond to novel situations beyond those used to actually train the network.

The success of the training effort is assessed by monitoring the degree of error between system responses and training examples as training progresses. Once this error is reduced beyond some specified threshold, we consider training complete. The fully

trained network is embedded into a program that provides a graphical user interface (GUI) for ease of application. This example illustrates the fact that the neuro-fuzzy synthesis offers us the ability to construct powerful, realistic experts systems cheaply and quickly.

CONCLUSIONS

Decision making in risk manage- ment is subject to two types of uncer- tainty. One stems from randomness, and is specified using probability. The other, often less recognized, is the un- certainty that results from knowledge imperfection. This uncertainty, known also as fuzziness, can be formalized using membership functions. These functions show the approximate rela- tionships between fuzzy concepts and the numerical properties of the real world.

The fuzzy reasoning process can also be computerized, using expert sys- tems. Automation can provide us a deeper understanding of the process, along with a platform for the wide- spread dissemination of expert knowl-

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edge. Recent advances in artificial neu- ral networks enhance our ability to deploy useful expert systems. Fuzzy expert systems provide us with realis- tic decision aids that pay proper respect to expert thought in risk management and insurance.

Suggested Reading

A good introduction to the various paradigms of uncertainty is Michael Smithson’s Ignorance and Uncer- tainty (Springer-Verlag, 1989).

For more information on the for- mal analysis of fuzziness, see Fuzzy Sets, Uncertainty and Information (Prentice Hall. 1988), authored by George Klir and Tina Folger.

A more complete description of the Retention Evaluation Program appears in the author’s “Retentions: An Expert Systems Approach” found in Risk Management, March, 1997.

Nikola Kasabov provides a com- plete treatment of expert knowledge representation using fuzzy, neural and neuro-fuzzy systems in Foundations of Neural Networks, Fuzzy Systems and Knowledge Engineering (MIT Press, 1996). The computer programs described in Dr. Kasabov’s book are available for download via the Internet at the University of Otago (New Zealand) Department of Information Sciences Web site: divcom.otago.ac.nz: L100/COMIINFOSCI/KEL/ fuzzycop.htm.